In Western use, mostly for blowing one’s nose or wiping sweat off one’s brow. Sometimes in a cinematic manner which evokes refinement in the course of labor. Might also be used to blot tears, either one’s own or to give to someone who has tears but no hanky.
In East Asian use, as a hand towel, since many public bathrooms do not provision paper towels. Very useful in the USA as well, since many restrooms are moving towards air dryers of varying quality. But a handkerchief is always on-hand and can also reduce time spent in the restroom.
I personally keep two handkerchiefs on me, for both the Western and East Asian use-cases.
The other comments have covered a lot of the background and variances throughout the world. But what I’ll add is that few countries are purely in one camp or the other. To use the USA as an example, criminal cases are adversarial, in the sense that the defense attorney will duke it out with the government’s attorney whether someone goes to prison.
For civil cases like a contract dispute, the procedure is closer to an inquisition system, although with the judge still merely presiding over the process. But attorneys in a USA civil case can depose witnesses, much like how (I think) a European judge-led inquisition would call a witness, and similar to how British coroners conduct an inquest (if murder mystery depictions on the BBC are accurate).
Perhaps the full thrust of the inquisition style can be found in USA federal agencies, whose rulemaking capacity requires asking direct questions to subject matter experts in a public forum, one which eventually leads to a determination on some germane topic, often enacting secondary legislation at the same time. Americans might not necessarily call such an action as a “ruling”, but evidence was taken, all sides were heard, and even public comment was accepted, before rendering a decision.
That said, one could argue that such “Article III” rulemaking (eg FCC Commissioners) or judgements (eg Immigration Court) are distinct from the traditional judicial rulings from “Article I” courts (eg US Supreme Court). But that’s a Constitutional wrinkle for another discussion.